cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A153977 One-fourth of partial sums of A153976.

Original entry on oeis.org

2, 9, 27, 65, 135, 252, 434, 702, 1080, 1595, 2277, 3159, 4277, 5670, 7380, 9452, 11934, 14877, 18335, 22365, 27027, 32384, 38502, 45450, 53300, 62127, 72009, 83027, 95265, 108810, 123752, 140184, 158202, 177905, 199395, 222777, 248159, 275652, 305370, 337430
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

Links

Crossrefs

Cf. A000217, A000537, A000578, A003215, A005898, A006007, A027602, A153976.

Programs

  • Maple
    A153977:=n->(1/4)*sum(i^3 + (i+2)^3, i=0..n): seq(A153977(n), n=0..50); # Wesley Ivan Hurt, Feb 04 2017
  • Mathematica
    a[n_]:=n^3;lst={};s=0;Do[s+=(a[n]+a[n+2]);AppendTo[lst,s/4],{n,0,6!}];lst
Accumulate[Array[#^3+(#+2)^3&,40,0]]/4 (* or *) LinearRecurrence[ {5,-10,10,-5,1},{2,9,27,65,135},40] (* Harvey P. Dale, Aug 02 2011 *)
  • PARI
    a(n)=(n^4 + 2*n^3 + 7*n^2 + 6*n)/8 \\ Charles R Greathouse IV, Feb 06 2017

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5, a(1)=2, a(2)=9, a(3)=27, a(4)=65, a(5)=135. - Harvey P. Dale, Aug 02 2011
a(n) = (A000217(n-1)^2 + A000217(n+1)^2 - 1)/4. - Richard R. Forberg, Dec 25 2013
Recurrence: (n-1)*(n^2 - n + 6)*a(n) = (n+1)*(n^2 + n + 6)*a(n-1). - Vaclav Kotesovec, Dec 26 2013
a(n) = A000217(A000217(n)) + A000217(n). - Bruno Berselli, May 28 2015
a(n) = (A000217(n)^2 + 3*A000217(n))/2 where A000217(n) is the n-th triangular number. - Frederic Isenmann, Feb 04 2017
Sum_{n>=1} 1/a(n) = 14/9 - 4*tanh(sqrt(23)*Pi/2)*Pi/(3*sqrt(23)). - Amiram Eldar, Aug 23 2022
From Elmo R. Oliveira, Aug 28 2025: (Start)
G.f.: x*(2 - x + 2*x^2)/(1-x)^5.
E.g.f.: x*(2 + x)^2*(4 + x)*exp(x)/8. (End)

A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12.

Original entry on oeis.org

0, 2, 14, 50, 130, 280, 532, 924, 1500, 2310, 3410, 4862, 6734, 9100, 12040, 15640, 19992, 25194, 31350, 38570, 46970, 56672, 67804, 80500, 94900, 111150, 129402, 149814, 172550, 197780, 225680, 256432, 290224, 327250, 367710, 411810, 459762
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

Comments

Partial sums of A011379.
Antidiagonal sums of the convolution array A213819. - Clark Kimberling, Jul 04 2012

Links

Crossrefs

Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819.

Programs

  • Mathematica
    With[{r=Range[0,50]},Accumulate[r^2+r^3]] (* Harvey P. Dale, Jan 16 2011 *)
Rest[CoefficientList[Series[-2 x^2 * (2 x + 1)/(x - 1)^5, {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 30 2014 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,2,14,50,130}, 25] (* G. C. Greubel, Sep 01 2016 *)
  • PARI
    concat(0, Vec(-2*x^2*(2*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Jun 28 2014
  • PARI
    a(n) = n*(n-1)*(n+1)*(3*n-2)/12 \\ Charles R Greathouse IV, Sep 01 2016

Formula

a(n) = 2 * A001296(n-1) = (n-1)*n*(n+1)*(3*n-2)/12 (n>0). - Bruno Berselli, Apr 21 2010
a(n) = Sum_{i=1..n-1} binomial(i+1,i)*i^2. - Enrique Pérez Herrero, Jun 28 2014
G.f.: 2*x^2*(2*x+1) / (1 - x)^5. - Colin Barker, Jun 28 2014
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - Vincenzo Librandi, Jun 30 2014
a(n) = Sum_{k=1..n-1}k*((n-1)*n/2 + k) for n > 1. - J. M. Bergot, Feb 16 2018
From Amiram Eldar, Aug 23 2022: (Start)
Sum_{n>=2} 1/a(n) = 141/5 - 9*sqrt(3)*Pi/5 - 81*log(3)/5.
Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 + 48*log(2)/5 - 129/5. (End)

Extensions

Edited by Bruno Berselli, Jun 15 2010
Simpler definition as suggested by Wesley Ivan Hurt, Jun 29 2014

A271636 a(n) = 4*n*(4*n^2 + 3).

Original entry on oeis.org

0, 28, 152, 468, 1072, 2060, 3528, 5572, 8288, 11772, 16120, 21428, 27792, 35308, 44072, 54180, 65728, 78812, 93528, 109972, 128240, 148428, 170632, 194948, 221472, 250300, 281528, 315252, 351568, 390572, 432360, 477028, 524672, 575388, 629272, 686420
Offset: 0

Views

Author

Vincenzo Librandi, Apr 11 2016

Keywords

Comments

This is the case h=0 of the identity 4*n*(4*n^2 + 3*(2*h + 1)^2) = (2*n - 2*h - 1)^3 + (2*n + 2*h + 1)^3.
Subsequence of A004999 and, after 0, second bisection of A153976.

Links

Crossrefs

Cf. A004999, A153976, A229183.

Programs

  • Magma
    [4*n*(4*n^2+3): n in [0..50]];
  • Mathematica
    Table[4 n (4 n^2 + 3), {n, 0, 50}]
  • PARI
    x='x+O('x^99); concat(0, Vec(x*(28+40*x+28*x^2)/(1-x)^4)) \\ Altug Alkan, Apr 11 2016
  • Python
    for n in range(0,1000):print(4*n*(4*n**2+3)) # Soumil Mandal, Apr 11 2016

Formula

O.g.f.: 4*x*(7 + 10*x + 7*x^2)/(1 - x)^4.
E.g.f.: 4*x*(7 + 12*x + 4*x^2)*exp(x). - Ilya Gutkovskiy, Apr 11 2016
a(n) = -a(-n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 4*A229183(2*n). - Bruno Berselli, Apr 11 2016

Extensions

Edit and extended by Bruno Berselli, Apr 12 2016

A272910 Numbers n such that (n-1)^3 + (n+1)^3 is a taxi-cab number (A001235).

Original entry on oeis.org

19, 32, 93, 124, 208, 243, 308, 395, 427, 471, 603, 672, 1057, 1568, 1892, 2181, 2223, 2587, 3040, 3049, 4037, 4336, 5232, 5556, 6196, 6305, 6643, 8288, 8748, 10161, 10185, 10612, 10985, 12352, 13741, 14807, 16021, 17568, 20352, 20653, 24080, 27216, 27867, 31113, 31869, 32032, 32500, 36593
Offset: 1

Views

Author

Altug Alkan, May 09 2016

Keywords

Comments

Numbers n such that 2*n*(n^2+3) is a member of A001235.
19 and 3049 are the only prime numbers in this sequence for n < 10^5.
How is the graph of second differences of this sequence?

Examples

			19 is a term because 18^3 + 20^3 = 13832 = 2^3 + 24^3.

Links

Crossrefs

Cf. A001235, A153976, A259836.

Programs

  • PARI
    T = thueinit(x^3+1, 1);
isA001235(n) = my(v=thue(T, n)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1;
lista(nn) = for(n=1, nn, if(isA001235(2*n*(n^2+3)), print1(n, ", ")));
Showing 1-4 of 4 results.

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